Equilibrium density calculation of generalized Muttalib-Borodin ensembles.

The Muttalib-Borodin (MB) random matrix ensemble, which has an additional interaction, was introduced as a solvable toy model for quasi one-dimensional (1D) disordered conductors. We generalize the MB random matrix ensemble with a disorder dependent parameter γ which characterizes the strength of the additional interaction term. This generalization can be considered as a simple toy model which captures the essential features of a quasi 1D to 3D generalization of disordered conductors and we call it the γ -ensembles. Our results suggest that the γ-ensembles can be mapped on to an MB ensemble by replacing the single particle confining potential with a γ-dependent effective confining potential. We numerically solve the Riemann-Hilbert (RH) problem associated with the equilibrium density of γ-ensembles for a range of γ between 0 and 1. We also study some interesting limits of the parameters of the γ-ensembles, where the equilibrium density for a generalized form of the β-ensembles can be explored.